So, we have here again we know as I said 400 nanometers to 700 nanometers or which comes to the energy of 1.8 electron volts for the 700 nanometers radiation and 3.1 electron volts for the 400-nanometer radiation. Now, something is exciting here we have a bandgap E g. So when radiation is incident on the material if the energy of the radiation is higher than the bandgap, then it enables a transition of an electron from the filled valence band into the conduction band. So for example, if I say let’s just use this value here we have a value 2 here, 2 electron volts okay. So, 2 electron volts I just assume that I have got a semiconductor with a bandgap of 2 electron volts and let us say since we have this range of radiation I will take the 2 extremes of the radiation I will assume that I have photons arriving on the sample which have 3.1 electron volts energy and some other photons arriving at the sample which are 1.8 electron volts energy. So, let’s say we have these options available to us. What will happen? Interestingly what happens is the higher energy radiation which is 3.1 electron volt radiation gets absorbed by the material because it is possible to do this transition that I just showed you it can do. So, even though this is only 2 electron volts, the 3.1 moves the electron in into the conduction band because it has got lot more energy pushes the electron into the conduction band it takes it above you know the starting point of the band. But and then maybe some other things will happen let us not worry about what happens at that point, but it will take an electron into the conduction band. On the other hand, the 1.8 electron volt photon that arrives at this material is unable to provide enough energy to clear this gap and therefore, the 1.8 electron volt photon is unable to do the transition. So, then what happens to the 1.8 electron volt radiation so, in fact, these materials end up being transparent to that radiation okay. So, generally speaking, they are transparent to radiation of wavelengths less than the bandgap and therefore, interestingly normally you think that you know some low energy thing will get absorbed high energy thing will be able to you know to pass through that’s you know the intuitive way we think of things. Here it is the other way around the high energy photon gets absorbed because it can cross the bandgap, but the low energy photon is unable to cross the bandgap. So, it just passes through the material without doing any transition. So it is, so, the material sort of is transparent to this radiation. So, therefore in fact, if the insulator were let’s say, for example, let’s say the insulator was just by looking at this let’s say its let I will just give a value of say 4.5 electron volts. So, 4.5 electron volts let us say the insulator has a bandgap of 4.5 electron volts then it will be completely transparent to the entire visible spectrum. So in fact, ceramic materials which are you know of high bandgap materials if you make them defect-free if you make them you know single crystal that defects free ceramic material. So, for example, zirconia if you take and you make a single crystal of zirconia that is defect-free you will find that it is clear crystal clear you can look through it and it will be transparent. Transparent meaning the entire visible radiation can go through it without interacting with the material because then at that point its bandgap will be higher than the value of the energies that are available in the visible part of the spectrum okay. So, this is how it works concerning these materials. Now, what would happen concerning a metal? Concerning a metal since you have a continuous set of energy levels there is no gap here, you have a continuous set of energy levels that are available just above this highest occupied energy level. So, therefore, for metals, any radiation that is incident on the incident on it can absorb and that is the reason why any metallic surface is opaque because no light goes through it no visible radiation goes through it. Essentially no other radiation will go through or the values of energy levels that are just above it are so closely spaced that essentially it is a continuous band for all practical purposes it is a continuous band any radiation incident on it will get absorbed. So, that is the idea here. But we have two things to do here if you use a metallic surface it gets absorbed heat gets absorbed and that is why we are using metallic surfaces for you know capturing the sunlight to pick up the heat and then use that heat or heating water or air or generating steam or a variety of things that we previously saw. But if you want to generate electricity, you have to absorb the sunlight you have to get these charge carriers and you have to separate the charge carriers and then use those charge carriers in some external circuit. So, you have some process that is involved in and for that, the metal then is not particularly useful to us because it just absorbs and it is all the electrons are all right there and you are not able to capture it as a separate electron that you can move into an external circuit at least not in this general layout. I mean a semiconductor there are ways in which you can know now pull off the electron and then take it to an external circuit. So, as we look at this photovoltaic approach to dealing with solar radiation through our next few classes that is the idea that we will look at. How is it that we can use semiconductors to create the situation where you have these electrons that can be pulled off into the external circuit and then used in you know as a power supply or a power source there which can then run something usefully? So, for that, we need to understand all this idea of you know how these bands come about, where is this, what is happening is there some more detail in this band structure that we have not fully captured are various materials with is the same bandgap the same in terms of how they handle the solar radiation. The answer is no there is some further finer detail here which decides which will tell you that some materials are better suited for solar cell application for photovoltaic applications than some other materials because there is some finer detail in the band structure which helps them in that idea. So, that’s sort of the thing that we will look at in this class. (Refer Slide Time: 25:23) So, we spoke about pure semiconductors and that is what we have listed here as intrinsic semiconductors. So, here there is no doping and it’s you know pure material and so there then we as I said the Fermi energy level E f is exactly halfway between the conduction band and the valence band and that’s how it is defined. Now when you do any kind of doping n-type doping to a semiconductor or p-type doping to a semiconductor you are essentially introducing an element into the material which either has an additional electron which can which it can put into the system or it has one less electron that it can put into the system relative to the base material. So, you can add dopants which you push we increase the number of electrons in the system or decrease the number of electrons in the system on average. So, those that increase their number of electrons are referred to as the and create this semiconductor referred to as the n-type semiconductor, n being you know indicating the fact that it’s now having this negative orientation with the negative charges and what it creates is, it creates set of donor levels which are very close to this conduction band. So, whereas, previously you had to if this were a pure semiconductor an intrinsic semiconductor you would have to move from here to here, to push the electron from the valence band to the conduction band in the n-type semiconductor you can you only have to do this small jump and you can get some you know conducting electrons. But that is only a limited number of conducting electrons if you want a large number of conducting electrons, you still have to go down to the valence band and pull it up, but you get this limited set of electrons which are close to the conduction band which can get into the conduction band, and that gives the semiconductor some interesting properties and that is why we look at us more often work with extrinsic semiconductors than we do with intrinsic semiconductors. Analogous to that and I know you know in sort of a similar, but opposite manner we have the p-type semiconductor here you are putting in materials as dopants which have one less electron than the base material that is present. So, it creates a bunch of acceptor levels. So, here again, the electrons which are on the top of this valence band can go into the acceptor level by just doing this small jump and it enables the material to start functioning in some interesting ways. So, this is the p-type extrinsic semiconductor and that’s the n-type. For the concerning the Fermi energy because it is you know integrated or intricately linked to the energy level of the electrons and the fact that that is where they are in a position to interact with their surroundings the Fermi energy level for an n-type semiconductor is essentially the very much the donor level energy level that is present there and the Fermi energy level for the p-type semiconductor is the acceptor level. So, this is how the Fermi energy level is certified. And as I said this is important for us to understand this definition for the Fermi energy level and also the location of this Fermi energy level in the energy scale because that will tell you what will happen when you put these materials in contact and that is something that we are going to see subsequently. So, that is the reason why I am drawing your attention to Fermi energy level now we will use it progressively. (Refer Slide Time: 28:46) So, what happens when you look at these intrinsic and extrinsic semiconductors? Just to give you an idea of how they behave differently when you take both the semiconductors and then you look at the charge carrier concentration which is what is marked on your y-axis here and you look at it as a function of temperature. At very close to 0 Kelvin all the charges are held within you know their base energy level their lowest energy level which could be you know the filled valence band or the acceptor level or the donor level it just sits there I mean nothing moves it's just sitting at those values. So, only when you start heating the material a little bit that the charges get a chance to start moving. Now, as I said because the n-type, as well as the p-type semiconductors, need a very small amount of energy here as well as here to get those electrons into a position where they can move if you see here at relatively low temperatures this is a temperature scale relatively low temperatures for the semiconductors that are extrinsic. The charged carriers get into a position where they can start carrying charge and moving around and showing you some behaviour corresponding to the existence of those charge carriers. So, at fairly low temperatures and in fact, this is if you this dotted line that I am showing you here is there would be even below a room temperature. So so, by the time you reach room temperature, all those charge carriers are already in a position to participate in some process that you are trying to put them to use for. So, typically if you look at the charged carrier concentration is fairly low temperatures you start giving energy and those charged carriers get into a position where they can start carrying charge and so they show up as part of the charged carrier concentration and then pretty quickly they max out at some value. That value will depend on the doping concentration based on how much dopants you put you may have a higher value or lower value, but essentially at a value, at a temperature below room temperature itself all the charge carriers become available and so that corresponds to some fixed charged carrier concentration which is in exactly in line with the doping concentration. So, it will level often at that value. And from there on for a fairly large range of temperature, the charged carrier concentration will remain flat and this is the charge carrier at a concentration that you end up using for a variety of purposes when you are using an extrinsic semiconductor and so this is referred to as the extrinsic regime of the semiconductor. If you go to very high temperatures significantly higher than room temperature then if you see here in addition to this transition from the donor level or the valence band to the acceptor level in addition to the transition you will also enable transitions from the valence band to the conduction band. You will have this full transition also occurring in addition to the original transition that you did. So, that is why you start seeing this additional behaviour here for an extrinsic semiconductor which is additional charge carriers which are now coming from the valence band. If you had an intrinsic semiconductor that’s completely pure then you have no extrinsic regime you only have it just tracks the x-axis up to this point and from there on it starts showing you this intrinsic regime. So, it only shows you the intrinsic regime it e doesn’t show you any other regime it stays flat till the time and then shows you the intrinsic regime. Whereas, the extrinsic semiconductor which has dopants first shows you the extrinsic regime and it shows you the extrinsic regime for a wide range of temperatures which are of interest to us which is basically from below room temperature to a significant value above room temperature and only after that it starts showing you the intrinsic regime and so in fact, mostly we are operating in this temperature range. We tend to operate in this temperature to for most of the properties that we put the semiconductors to use for. So, this is something that you should keep in mind when you look at all these semiconductors. So, even though we show you, I show you this intrinsic semiconductor initially, and I compared it with you know insulators and metals, and I also showed you where the Fermi energy for an intrinsic semiconductor is it and how it compares with extrinsic semiconductors you will you generally find that much of the discussion is mostly on extrinsic semiconductors. We don’t use intrinsic semiconductors as much as we do with extending as we use extrinsic semiconductors which are the doped semiconductors because of this interest these interesting properties it provides to us by this extrinsic regime that is present with which we operate the material. (Refer Slide Time: 33:24) Now, in the next few slides are after this small calculation that I am going to show you the next few slides may look a little complicated, but I am just going to show you some few highlights which help you understand why there is additional detail in the band structure which is not been shown to you so far. But in the end, I will summarize it and the summary is what is more important for you to keep in mind, but I will walk you through a few slides which show you the detail before we reach that summary. So, once we start talking of electrons in solids over the years it has been recognized that it is more appropriate to use quantum mechanical principles to talk about electrons. So, we will not go into great detail of the quantum mechanics and how it is developed, but the most important equations which we are in fact, familiar from high school days is all I am going to use here, but this is relevant for the next few diagrams that I am going to show you and that is why I am going to develop them for you. So, Planck put this I mean was the first to discover this equation E equals h nu and that is considered as you know the equation that originated that led to that that defined the discovery of quantum mechanics. And then de Broglie extended this idea and said that you know if any particle has a momentum p then you can associate a wavelength with it called the de Broglie wavelength lambda which is given by as h by p and p is mv that is the momentum mv the momentum right. So, now, if you extend this a little bit and rearrange and extend things a little bit we have p equals h by lambda and then because of some notation that we use in physics instead of just using h the preference is to use h by 2 pis and similarly instead of just using lambda we use 2 pis by lambda you can see that if you multiply the 2 pis cancel out and it is the same as h by lambda okay. And this notation has some detail I mean names associated with it. So, this h by 2 pis is referred to as h bar, so in books in physics sometimes you will see h bar that’s what they are referring to this is simply h by 2 pi Planck’s constant divided by 2 pi and this 2 pi by lambda is referred to ask or the wave vector k equals 2 pis by lambda h bar equals h by 2 pi k equals 2 pis by lambda that’s it okay. So, this is called a wave vector okay. So, this is what it is. So, now, if you keep this these quantities in mind. So, if you look at energy and how it relates to momentum energy we write as half mv square right. So, if I want to relate this to momentum and momentum is m v, so I simply play around a little bit to the numbers, so I have simply half m square v square by m. I have just multiplied the numerator and denominator by m. So, m square v square is nothing, but p square because p equals mv right p square and p itself is h bark. So, instead of writing m square v square, I can write h bar square k square. So, I will write h bar square k square right because p equals mv and p is also equal to h bar kk and I have p square here. So, I write h bar square k square by 2 m. So, energy is h bar square k square by 2m and that’s the equation that I have got down here right. So, why is this interesting? So, this is an equation, this is the equation is related to the quantum mechanical approach of looking at electrons in solids. So, this is the equation this is here the h bar is a constant. So, it’s h by 2 pis is a constant, 2 is a constant of course, and m is the mass of the electron. So, that is also a constant. So, what do we have that is related here we simply have E here which is the energy and k here which is the wave vector which is 2 pi by lambda. So, when an electron moves with different energies you have a different wavelength associated with it based on the momentum that it has and for each wavelength that you have associated with the electron you can find out the energy associated with it using this equation right. So, now, if you, I am going to now, this if you see if you plot E versus k. It is going to be a parabola. So, k is a wave vector and it can have positive values or negative values which simply represents the direction in which the electron is moving. We will consider a one-dimensional case. So, it can move in the positive x-direction or in the positive or in the negative x-direction and so I simply have plus k or a minus k and regardless of whether it is plus k or minus k E will be k square proportional to k square right. So, if you put E proportional to k square you will get a parabola right. So, this is called the free-electron parabola it represents all the energy values that the electron can assume and the corresponding wavelength values that it will have right. So, in the next few plots, I am going to show you this parabola and then I am going to show you some distortions of the parabola and what is the significance of this distortion from the perspective of a band structure. So, that is the idea that I am going to pursue. So, you are going to see a few diagrams which show the parabola and then eventually the distortion of the parabola. There will be a few terms you may not be familiar with I will briefly tell you what those terms are as we go along. As I said at the end of those slides I will summarize it with one key result which is a result that is of immediate relevance to us. (Refer Slide Time: 39:08) Okay so, this blue line that you see here is that free electron parabola which is essentially E equals h bar square k square by 2 m. So, I have plotted E as a function of k, k is here and energy E is here okay and I have this is this k is in the 2 pis by lambda various values of 2 pis by lambda are plotted here okay. So, now, so that is the blue curve that you see there. So, the basic idea is that the electrons are travelling in this material and they have various velocities associated with them therefore, they have various wavelengths associated with them. Now, we know from general science that when you have radiation of some wavelength and it interacts with the material that has that it that has a periodic structure you have diffraction right. So, we realize that the typical metal or semiconductor that you have or an insulator that you have is typically crystalline material mostly we are discussing crystalline materials they have some periodic structure. So, when when you have periodic structure and you have electrons within that material travelling with some wavelength associated with them they can interact with that periodic structure. So, we will briefly look at that’s essentially what we are going to look at the next few slides. We will talk about this, I will briefly tell you what this Brillouin zone is. In this plot and in the subsequent plots that I am going to show you we are plotting 2 pieces of information. The first is this E equals h bar square by h bar square k square by 2 m which is that curve that you see and these dotted lines that you see represent that periodic structure of the material okay without and this a is from that it is you know if you assume a one-dimensional lattice of spacing a. So, you start at the origin at a there is a lattice point at 2 a there is another lattice point, 3 a that is a lattice point, etcetera, this a is that lattice spacing okay. So, it is simply since we are comparing it with 2 pi by lambda we are plotting here similarly a in the inverse notation okay. So, this is called reciprocal space notation not very critical for you to know that name. But the point is what is plotted on your x-axis at those specific values where I am plotting those dotted lines represent the periodic structure of the material and you can see all the parameters here which, therefore, indicate that. So, there is a periodic structure for the material which is represented by those dotted lines and there is a free electron parabola which corresponds to the behaviour of the electrons inside that material. In the few slides that we are going to see how this periodic structure interacts with this parabola. And this periodic structure has this name called Brillouin zones and that’s all you need to know about it if you need to know more about it there are courses on physics of materials which discuss this in greater detail. But it's sufficient for you to know that there is something called a periodic structure and one way to represent it is using this inverse notation where a gets represented as 2 pi by a and in that process, you get this set of Brillouin zones at a spacing of 2 pi by a. So, that’s all you need to know, but just remember there is periodicity here wave information here both of them are here. (Refer Slide Time: 42:13) So, we know from Bragg’s law that n lambda equals 2 d sin theta n lambda equals 2 d sin theta is one way in which we represent the diffraction condition in materials which is sort of what is shown in the plot here how it gets it derived. So, d is indicated here and you have some wavelength lambda coming in and d sine theta is the additional distance it travels here as well as the additional distance it travels here in which this wave here travels relative to this wave the additional distance that the second wave travels relative to the first wave. And if that spacing is an integral multiple of if the wavelength is an integral multiple of that 2 d sine theta then you have constructive interference right. So, that is how you get this Bragg law. (Refer Slide Time: 43:12) Now the same thing is also shown in another notation. It’s the same idea here where these dots represent the periodic structure of the material and we are looking at the same radiation in inverse notation. So, again you need not know great detail of it, but they accept that there is an equation here which represents the this S nought by lambda here represents your incident beam this S by lambda represents the potential diffracted beam, could be, could be not we don’t know. So, in that direction, we are looking for a diffracted beam. If it equals a reciprocal lattice vector for example, here I have OB it’s a reciprocal lattice vector. So, if it equals ob if the vector difference between these two equals this vector here then diffraction occurs. It is analogous to this equation here where you have wavelength information here, you have spacing information here right this is the same thing. In reciprocal notation, you have wavelength information in the S by lambda and S nought by lambda and you have spacing information in H hkl that’s all you need to know. So, what it means is that when you have specific wavelengths and those wavelengths result in this you know on touching of these reciprocal lattice points you get diffraction this is basically what the whole idea is. (Refer Slide Time: 44:29) Now, at the Brillouin zones, this is exactly what is happening. So, you have these boundaries these dotted line boundaries which represent this periodic structure that is present in the material. When a wavelength of the corresponding wavelength touches that boundary diffraction occurs, so at these boundaries diffraction wherever the free-electron parabola is touching these boundaries diffraction is the condition for diffraction is valid okay. So, what happens is this free electron parabola is represents as I said all the energies that the electron can possess all right. So, when it is free when it is not seeing a periodic structure when it is free to run around without any periodicity in the present in the material then it can assume all of those energy levels okay. So, there is no restriction on the energy level it can take. For every energy level, there is a corresponding wavelength that it can have and you can do whatever it is possible with that free electron parabola. When the same electron is put inside a structure that has a periodic lattice then it tries to do this free electron parabola it tries to assume all energy values that are consistent with that free electron parabola. So, E equals h bar square k square by 2 m. So, the electron tries to assume all values consistent with this E equals h bar square k square by 2 m where k is that 2 pi by lambda, but it finds that at values corresponding to pi by a corresponding to 2 pi by a when lambda equals pi by a lambda equals 2 pi by lambda equals 3 pi by etcetera. It’s meeting or rather 2 pi by lambda equals 2 pi by a, pi by a, 2 pi by a, 3 pi by a 1 E whenever it meets those conditions it is satisfying the condition for diffraction. Because of this diffraction occurring at those values of wavelengths it creates some standing waves and it creates a small gap there is a way to calculate it, but it creates a gap where it is unable to hold those energy values. So, that is the reason this free electron parabola which is the dotted line that you see here gets converted to a distorted version of it which is this green line that you see here. So, whereas, you previously had a dotted line going this way at the boundaries there is a distortion of the dotted line and you see this curve that you see here and that is basically what you see here and so on. So, now, what we have done, we suddenly find that this set of energy values here which are not allowed for the electron. This set of energy values that are not allowed for the electron is that bandgap that you see okay. So, only when you go into the structure of the material in this level of detail do you understand where is this bandgap coming from? Till now mostly you know simply heard that there is a bandgap you have heard that there is a valence band there is a conduction band in between there is a bandgap that’s all the information we have. But only when you get into this level of detail do you understand where this bandgap is coming from, what is that interaction that is happening in the material that creates the situation, what, there is a set of energy levels that are forbidden? So, this is the origin of the bands, band structure in the material. (Refer Slide Time: 47:29) And to be more specific whatever you see here which is the allowed set of values becomes an allowed band and whatever is not allowed here becomes this bandgap right. So, this is how you get and so in fact, the material you have many bands, you have a band right at the bottom then this a bandgap here, another band here, another band gap here and there is a band here. If these two were the last two bands that were present in the material then yes we would call the topmost band as the conduction band and the immediately lower band which is full would then be referred to as the valence band. So, this is how the material begins to show itself. And the one additional detail that we need to add here which will complete the picture is the presence of the Fermi energy is the highest energy level occupied by electrons. So, that could be anywhere. So, that is one value here. So, if this were the Fermi energy level up here it means that up to that electrons are full. So, this band is full here, this band is full here and this band on top is full up to this point and that is how you get the Fermi energy information also into this and you can see here and in this case, may have an example that I am showing you it is a situation where this is a metallic sample because t
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